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Matrices & Their Arithmetic

requiresSystems of Linear Equations ⚠Vectors ⚠

⚗ Dr. Möbius, from the lab

You spent several lessons turning systems of equations into geometry. Good. Now I'm going to show you what happens when you sand off all the variables, keep the numbers, and treat the resulting grid as a single mathematical object. The matrix. It stores everything — and we haven't even gotten to the transformation magic yet, that's next — but for now I need you to respect the grid. Touch the wrong index once in this lab and the whole beaker shatters.

THE BIG IDEA

A matrix is a rectangular array of numbers indexed by row and column; matrices support entrywise addition and scalar multiplication, and a transpose operation that flips rows and columns.

What a matrix is

A matrix is a rectangular grid of numbers. Here is a $2 \times 3$ matrix (two rows, three columns):

$A = \begin{pmatrix} 1 & -2 & 5 \\ 0 & 3 & -1 \end{pmatrix}$

The dimensions of a matrix are always stated rows-first, columns-second: an $m \times n$ matrix has $m$ rows and $n$ columns. The entry in row $i$ and column $j$ is written $a_{ij}$ or $A_{ij}$ or $(A)_{ij}$ — all the same, all slightly annoying. Row index first, column index second, always — this is not a convention you get to disagree with. Getting it backwards is the canonical beginner mistake, and if you do it in my lab you will write " $a_{ij}$ means row $i$ , column $j$ " on the blackboard until the chalk runs out and then you'll continue with your finger.

Single-column matrices ( $n = 1$ ) are column vectors — you've been using these as vectors since the previous two lessons. Single-row matrices ( $m = 1$ ) are row vectors. A $1 \times 1$ matrix is just a scalar with delusions of grandeur.

Three jobs matrices hold

Before we compute anything, let me tell you why matrices exist. They serve three distinct purposes in this course:

Storing a linear system's coefficients. The system $2x + 3y = 5$ , $x - y = 1$ becomes the matrix $\begin{pmatrix} 2 & 3 \\ 1 & -1 \end{pmatrix}$ with the $x$ 's and $y$ 's sandblasted off. All the information is there; nothing is lost. This is why you saw elimination in the Algebra stratum — it's secretly matrix row reduction, and you'll meet it again in a later lesson as Gaussian elimination.
Storing data in a table. A spreadsheet is a matrix. A grayscale image is a matrix (each entry is a pixel brightness). A social network's connections can be a matrix. This use is important in applications but not our main concern here.
Encoding a transformation. This is the real one. A $2 \times 2$ matrix isn't just a table of four numbers — it's a function that moves the entire plane around. This third use is the reason this stratum exists, and we will spend the next several lessons building up to it.

Addition and scalar multiplication: embarrassingly simple

Adding two matrices of the same size: add the corresponding entries.

$\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} + \begin{pmatrix} 5 & -1 \\ 0 & 7 \end{pmatrix} = \begin{pmatrix} 1+5 & 2+(-1) \\ 3+0 & 4+7 \end{pmatrix} = \begin{pmatrix} 6 & 1 \\ 3 & 11 \end{pmatrix}$

You cannot add matrices of different sizes. $\mathbb{R}^{2\times 2} + \mathbb{R}^{2\times 3}$ is undefined, exactly the way you can't add vectors of different lengths. Dimensions must agree.

Scalar multiplication: multiply every entry by the scalar.

$3 \begin{pmatrix} 1 & -2 \\ 5 & 0 \end{pmatrix} = \begin{pmatrix} 3 & -6 \\ 15 & 0 \end{pmatrix}$

All the familiar laws hold: commutativity and associativity of addition, distributivity of scalar multiplication over matrix addition. These are inherited component-by-component from the real number laws. I won't prove them because they're exercises, and I strongly — no, I furiously — suggest you verify at least one with a $2\times 2$ example. Seriously. Do the damn check. Nine seconds. Go.

The zero matrix $O$ (all entries zero) is the additive identity: $A + O = A$ .

Transpose: flipping rows and columns

The transpose of a matrix $A$ , written $A^T$ , is the matrix you get by reflecting $A$ across its main diagonal — row $i$ becomes column $i$ , or equivalently $(A^T)_{ij} = A_{ji}$ .

$A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} \quad \Longrightarrow \quad A^T = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

The $2 \times 3$ matrix became a $3 \times 2$ matrix. In general, the transpose of an $m \times n$ matrix is $n \times m$ .

A symmetric matrix satisfies $A = A^T$ , meaning $a_{ij} = a_{ji}$ for all $i, j$ . Symmetric matrices are necessarily square, and their entries mirror across the main diagonal:

$\begin{pmatrix} 1 & 3 \\ 3 & 7 \end{pmatrix} = \begin{pmatrix} 1 & 3 \\ 3 & 7 \end{pmatrix}^T$

Symmetric matrices are extremely special — they are the royalty of the matrix world and they know it. They'll headline the final boss of this entire course, the Spectral Theorem, which is so damn beautiful it should be illegal. File the word "symmetric" away and treat it with reverence whenever it appears.

The identity matrix

The identity matrix $I_n$ (or just $I$ when the size is clear) is the $n \times n$ matrix with $1$ 's on the main diagonal and $0$ 's everywhere else:

$I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}, \quad I_3 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}$

The identity matrix will be the " $1$ " of matrix multiplication (next lesson). It's also symmetric, self-transpose, and has every column and row summing to exactly $1$ . It is, in a very real sense, the do-nothing transformation — it leaves every vector exactly where it was. This makes it the matrix version of the number $1$ , the "identity element" for multiplication. It is called the identity matrix for exactly that reason.

🔬 SPECIMENS (worked examples)

Worked example 1 — matrix addition and scalar multiplication▸

Let $A = \begin{pmatrix} 2 & -1 \\ 0 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 4 \\ -2 & 1 \end{pmatrix}$ . Compute $2A + B$ .

Step 1: Scalar multiplication first — multiply every entry of $A$ by $2$ : $2A = \begin{pmatrix} 4 & -2 \\ 0 & 10 \end{pmatrix}$

Step 2: Add entrywise: $2A + B = \begin{pmatrix} 4 & -2 \\ 0 & 10 \end{pmatrix} + \begin{pmatrix} 3 & 4 \\ -2 & 1 \end{pmatrix} = \begin{pmatrix} 4+3 & -2+4 \\ 0+(-2) & 10+1 \end{pmatrix} = \begin{pmatrix} 7 & 2 \\ -2 & 11 \end{pmatrix}$

Check each entry: $(1,1)$ : $4+3=7$ . $(1,2)$ : $-2+4=2$ . $(2,1)$ : $0-2=-2$ . $(2,2)$ : $10+1=11$ . All correct.

Worked example 2 — transpose and identifying symmetry▸

Compute $A^T$ for $A = \begin{pmatrix} 1 & 3 & -2 \\ 4 & 0 & 7 \end{pmatrix}$ . Then determine whether $B = \begin{pmatrix} 5 & -1 \\ -1 & 3 \end{pmatrix}$ is symmetric.

Transpose of $A$ : Row $1$ of $A$ becomes column $1$ of $A^T$ ; row $2$ becomes column $2$ . $A^T = \begin{pmatrix} 1 & 4 \\ 3 & 0 \\ -2 & 7 \end{pmatrix}$

$A$ was $2\times 3$ ; its transpose is $3\times 2$ .

Symmetry of $B$ : Compute $B^T$ by flipping the off-diagonal entries. $B^T = \begin{pmatrix} 5 & -1 \\ -1 & 3 \end{pmatrix}$

$B = B^T$ , so yes, $B$ is symmetric. Notice: the diagonal can be anything; the test is whether $b_{12} = b_{21}$ (and in larger matrices, $b_{ij} = b_{ji}$ for all off-diagonal pairs).

Worked example 3 — the trap: rows first, columns second, or suffer▸

For the matrix $M = \begin{pmatrix} 7 & 2 & -1 \\ 5 & 0 & 4 \end{pmatrix}$ , find the entries $m_{12}$ , $m_{21}$ , and $m_{23}$ .

Remember: $m_{ij}$ means row $i$ , column $j$ .

$m_{12}$ : row $1$ , column $2$ . Row $1$ is $(7, 2, -1)$ ; column $2$ of that is $\mathbf{2}$ .
$m_{21}$ : row $2$ , column $1$ . Row $2$ is $(5, 0, 4)$ ; column $1$ of that is $\mathbf{5}$ .
$m_{23}$ : row $2$ , column $3$ . Row $2$ is $(5, 0, 4)$ ; column $3$ of that is $\mathbf{4}$ .

The common trap is to read $m_{21}$ as "row 2, column 1 = 0" because you scanned the second entry of the second row. But column 1 of row 2 is the first entry of the second row, which is $5$ . Slow down and count.

☠ KNOWN HAZARDS

Reading the index backwards. $a_{ij}$ means row $i$ , column $j$ — NOT the other way around. The row index comes first in every convention, everywhere, forever. If you write $a_{32}$ when you mean row 2, column 3, every single matrix computation you do will be wrong and you will go insane trying to figure out why.
Adding matrices of different sizes. Matrix addition is only defined when the dimensions match. $A + B$ requires $A$ and $B$ to have exactly the same number of rows AND the same number of columns.
Thinking $(A^T)^T =$ something new. Transposing twice returns you to the original: $(A^T)^T = A$ . It's a flip; two flips is where you started.
Thinking scalar multiplication changes the dimensions. Multiplying a $3\times 2$ matrix by a scalar gives a $3\times 2$ matrix. The size never changes under scalar multiplication — only under operations like transposition or (later) matrix multiplication.

TL;DR

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A matrix is an $m\times n$ grid of numbers; entry $a_{ij}$ is in row $i$ and column $j$ — rows first, always.
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Addition and scalar multiplication work entrywise; both matrices must have the same dimensions for addition to be defined.
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Transpose: $(A^T)_{ij} = A_{ji}$ . Flips rows into columns; turns an $m\times n$ matrix into $n\times m$ .
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Symmetric matrices satisfy $A = A^T$ ; they are always square and will become very important in later strata.
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The identity matrix $I_n$ is the $n\times n$ matrix with $1$ 's on the diagonal and $0$ 's elsewhere — the do-nothing matrix.

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